Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements

Abstract

We develop a hypergraph-theoretic framework for Spekkens contextuality applied to Kochen-Specker (KS) type scenarios that goes beyond the Cabello-Severini-Winter (CSW) framework. To do this, we add new hypergraph-theoretic ingredients to the CSW framework. We then obtain noise-robust noncontextuality inequalities in this generalized framework by applying the assumption of (Spekkens) noncontextuality to both preparations and measurements. The resulting framework goes beyond the CSW framework in both senses, conceptual and technical. On the conceptual level: 1) as in any treatment based on the generalized notion of noncontextuality à la Spekkens, we relax the assumption of outcome determinism inherent to the Kochen-Specker theorem but retain measurement noncontextuality, besides introducing preparation noncontextuality, 2) we do not require the $\textit{exclusivity }$ $\textit{principle}$ -- that pairwise exclusive measurement events must all be mutually exclusive -- as a fundamental constraint on measurement events of interest in an experimental test of contextuality, given that this property is not true of general quantum measurements, and 3) as a result, we do not need to presume that measurement events of interest are ``sharp" (for any definition of sharpness), where this notion of sharpness is meant to imply the exclusivity principle. On the technical level, we go beyond the CSW framework in the following senses: 1) we introduce a source events hypergraph -- besides the measurement events hypergraph usually considered -- and define a new operational quantity ${\rm Corr}$ that appears in our inequalities, 2) we define a new hypergraph invariant -- the $\textit{weighted}$ $\textit{max-predictability}$ -- that is necessary for our analysis and appears in our inequalities, and 3) our noise-robust noncontextuality inequalities quantify tradeoff relations between three operational quantities -- ${\rm Corr}$, $R$, and $p_0$ -- only one of which (namely, $R$) corresponds to the Bell-Kochen-Specker functionals appearing in the CSW framework; when ${\rm Corr}=1$, the inequalities formally reduce to CSW type bounds on $R$. Along the way, we also consider in detail the scope of our framework vis-à-vis the CSW framework, particularly the role of Specker's principle in the CSW framework, i.e., what the principle means for an operational theory satisfying it and why we don't impose it in our framework.